Second-order Stationary Points Research Articles

Nonconvex Factorization and Manifold Formulations Are Almost Equivalent in Low-Rank Matrix Optimization

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish a sandwich relation on the spectrum of Riemannian and Euclidean Hessians at first-order stationary points (FOSPs). As a result of that, we obtain an equivalence on the set of FOSPs, second-order stationary points, and strict saddles between the manifold and factorization formulations. In addition, we show that the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another. Similarities and differences in the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and factorization formulations for handling rank constraints, and it provides a geometric explanation for the similar empirical performance of factorization and manifold approaches in low-rank matrix optimization observed in the literature. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections (in particular, the sandwich relation), we are able to solve unanswered questions in the literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing. Funding: This work was supported by the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107 and 62141407], and the Shanghai Science and Technology Program [Grant 21JC1400600]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0030 .

Quantization avoids saddle points in distributed optimization

Distributed nonconvex optimization underpins key functionalities of numerous distributed systems, ranging from power systems, smart buildings, cooperative robots, vehicle networks to sensor networks. Recently, it has also merged as a promising solution to handle the enormous growth in data and model sizes in deep learning. A fundamental problem in distributed nonconvex optimization is avoiding convergence to saddle points, which significantly degrade optimization accuracy. We find that the process of quantization, which is necessary for all digital communications, can be exploited to enable saddle-point avoidance. More specifically, we propose a stochastic quantization scheme and prove that it can effectively escape saddle points and ensure convergence to a second-order stationary point in distributed nonconvex optimization. With an easily adjustable quantization granularity, the approach allows a user to control the number of bits sent per iteration and, hence, to aggressively reduce the communication overhead. Numerical experimental results using distributed optimization and learning problems on benchmark datasets confirm the effectiveness of the approach.

Trust Region Methods for Nonconvex Stochastic Optimization beyond Lipschitz Smoothness

In many important machine learning applications, the standard assumption of having a globally Lipschitz continuous gradient may fail to hold. This paper delves into a more general (L0, L1)-smoothness setting, which gains particular significance within the realms of deep neural networks and distributionally robust optimization (DRO). We demonstrate the significant advantage of trust region methods for stochastic nonconvex optimization under such generalized smoothness assumption. We show that first-order trust region methods can recover the normalized and clipped stochastic gradient as special cases and then provide a unified analysis to show their convergence to first-order stationary conditions. Motivated by the important application of DRO, we propose a generalized high-order smoothness condition, under which second-order trust region methods can achieve a complexity of O(epsilon(-3.5)) for convergence to second-order stationary points. By incorporating variance reduction, the second-order trust region method obtains an even better complexity of O(epsilon(-3)), matching the optimal bound for standard smooth optimization. To our best knowledge, this is the first work to show convergence beyond the first-order stationary condition for generalized smooth optimization. Preliminary experiments show that our proposed algorithms perform favorably compared with existing methods.

A Second-Order Finite-Difference Method for Derivative-Free Optimization

In this paper, a second-order finite-difference method is proposed for finding the second-order stationary point of derivative-free nonconvex unconstrained optimization problems. The forward-difference or the central-difference technique is used to approximate the gradient and Hessian matrix of objective function, respectively. The traditional trust-region framework is used, and we minimize the approximation trust region subproblem to obtain the search direction. The global convergence of the algorithm is given without the fully quadratic assumption. Numerical results show the effectiveness of the algorithm using the forward-difference and central-difference approximations.

Escaping saddle points efficiently with occupation-time-adapted perturbations

Motivated by the super-diffusivity of self-repelling random walk, which has roots in statistical physics, this paper develops a new perturbation mechanism for optimization algorithms. In this mechanism, perturbations are adapted to the history of states via the notion of occupation time. After integrating this mechanism into the framework of perturbed gradient descent (PGD) and perturbed accelerated gradient descent (PAGD), two new algorithms are proposed: perturbed gradient descent adapted to occupation time (PGDOT) and its accelerated version (PAGDOT). PGDOT and PAGDOT are guaranteed to avoid getting stuck at non-degenerate saddle points, and are shown to converge to second-order stationary points at least as fast as PGD and PAGD, respectively. The theoretical analysis is corroborated by empirical studies in which the new algorithms consistently escape saddle points and outperform not only their counterparts, PGD and PAGD, but also other popular alternatives including stochastic gradient descent, Adam, and several state-of-the-art adaptive gradient methods.

A Generalized Nesterov-Accelerated Second-Order Latent Factor Model for High-Dimensional and Incomplete Data.

High-dimensional and incomplete (HDI) data are frequently encountered in big date-related applications for describing restricted observed interactions among large node sets. How to perform accurate and efficient representation learning on such HDI data is a hot yet thorny issue. A latent factor (LF) model has proven to be efficient in addressing it. However, the objective function of an LF model is nonconvex. Commonly adopted first-order methods cannot approach its second-order stationary point, thereby resulting in accuracy loss. On the other hand, traditional second-order methods are impractical for LF models since they suffer from high computational costs due to the required operations on the objective's huge Hessian matrix. In order to address this issue, this study proposes a generalized Nesterov-accelerated second-order LF (GNSLF) model that integrates twofold conceptions: 1) acquiring proper second-order step efficiently by adopting a Hessian-vector algorithm and 2) embedding the second-order step into a generalized Nesterov's acceleration (GNA) method for speeding up its linear search process. The analysis focuses on the local convergence for GNSLF's nonconvex cost function instead of the global convergence has been taken; its local convergence properties have been provided with theoretical proofs. Experimental results on six HDI data cases demonstrate that GNSLF performs better than state-of-the-art LF models in accuracy for missing data estimation with high efficiency, i.e., a second-order model can be accelerated by incorporating GNA without accuracy loss.

Second-order optimality conditions for interval-valued functions

This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of second-order optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the first-order conditions. We will define new concepts such as second-order gH-derivative for interval-valued functions, 2-critical points, and 2-KKT-critical points. We obtain and present new types of interval-valued functions, such as 2-pseudoinvex, characterized by the property that all their second-order stationary points are global minima. We extend the optimality criteria to the semi-infinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with interval-valued functions.

A Newton-CG Based Augmented Lagrangian Method for Finding a Second-Order Stationary Point of Nonconvex Equality Constrained Optimization with Complexity Guarantees

A Newton-CG Based Augmented Lagrangian Method for Finding a Second-Order Stationary Point of Nonconvex Equality Constrained Optimization with Complexity Guarantees

Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

A Newton-CG Based Barrier Method for Finding a Second-Order Stationary Point of Nonconvex Conic Optimization with Complexity Guarantees

A Newton-CG Based Barrier Method for Finding a Second-Order Stationary Point of Nonconvex Conic Optimization with Complexity Guarantees

On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-Rank Matrix Optimization

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization, and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients—including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries—are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature. Funding: X. Li was partially supported by National Key R&D Program of China [Grants 2020YFA0711900, 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107, 62141407], the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], the “Chenguang Program” by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission [Grant 19CG02], the Shanghai Science and Technology Program [21JC1400600]. Y. Luo and A. R. Zhang were partially supported by the National Science Foundation [CAREER-2203741].

Dissolving Constraints for Riemannian Optimization

In this paper, we consider optimization problems over closed embedded submanifolds of [Formula: see text], which are defined by the constraints c(x) = 0. We propose a class of constraint-dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint-dissolving function ( CDF ). Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint-dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint-dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint-dissolving approaches. Funding: The research of N. Xiao and K.-C. Toh is supported by the Ministry of Education of Singapore Academic Research Fund Tier 3 [Grant MOE-2019-T3-1-010]. The research of X. Liu is supported in part by the National Natural Science Foundation of China [Grants 12125108, 11971466, 12288201, 12021001, and 11991021]; the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711904]; and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences [Grant ZDBS-LY-7022].

A Momentum-Accelerated Hessian-Vector-Based Latent Factor Analysis Model

Service-oriented applications commonly involve high-dimensional and sparse (HiDS) interactions among users and service-related entities, e.g., user-item interactions from a personalized recommendation services system. How to perform precise and efficient representation learning on such HiDS interactions data is a hot yet thorny issue. An efficient approach to it is latent factor analysis (LFA), which commonly depends on large-scale non-convex optimization. Hence, it is vital to implement an LFA model able to approximate second-order stationary points efficiently for enhancing its representation learning ability. However, existing second-order LFA models suffer from high computational cost, which significantly reduces its practicability. To address this issue, this paper presents a Momentum-accelerated Hessian-vector algorithm (MH) for precise and efficient LFA on HiDS data. Its main ideas are two-fold: a) adopting the principle of a Hessian-vector-product-based method to utilize the second-order information without manipulating a Hessian matrix directly, and b) incorporating a generalized momentum method into its parameter learning scheme for accelerating its convergence rate to a stationary point. Experimental results on nine industrial datasets demonstrate that compared with state-of-the-art LFA models, an MH-based LFA model achieves gains in both accuracy and convergence rate. These positive outcomes also indicate that a generalized momentum method is compatible with the algorithms, e.g., a second-order algorithm, which implicitly rely on gradients.

Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of $$O(\epsilon ^{-3/2})$$ to obtain an $$(\epsilon ,\sqrt{\epsilon })$$ -second-order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $$\epsilon $$ and minimum eigenvalue of Riemannian Hessian lower bounded by $$-\sqrt{\epsilon }$$ , is established when the manifold is embedded in the Euclidean space. Furthermore, the paper proposes a computationally more appealing modification of the algorithm which only requires an inexact solution of the cubic regularized Newton subproblem with the same iteration complexity. The proposed algorithm is evaluated and compared with three other Riemannian second-order methods over two numerical studies on estimating the inverse scale matrix of the multivariate t-distribution on the manifold of symmetric positive definite matrices and estimating the parameter of a linear classifier on the sphere manifold.

Second-Order Guarantees of Stochastic Gradient Descent in Nonconvex Optimization

Recent years have seen increased interest in performance guarantees of gradient descent algorithms for nonconvex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary points. Most available works limit the gradient noise component to be bounded with probability one or sub-Gaussian and leverage concentration inequalities to arrive at high-probability results. We present an alternate approach, relying primarily on mean-square arguments and show that a more relaxed relative bound on the gradient noise variance is sufficient to ensure efficient escape from saddle points without the need to inject additional noise, employ alternating step sizes, or rely on a global dispersive noise assumption, as long as a gradient noise component is present in a descent direction for every saddle point.

Stochastic cubic-regularized policy gradient method

Policy-based reinforcement learning methods have achieved great achievements in real-world decision-making problems. However, the theoretical understanding of policy-based methods is still limited. Specifically, existing works mainly focus on first-order stationary point policies (FOSPs); in some very special reinforcement learning settings (e.g., tabular case and function approximation with restricted parametric policy classes) some works consider globally optimal policy. It is well-known that FOSPs could be undesirable local optima or saddle points, and obtaining a global optimum is generally NP-hard. In this paper, we propose a policy gradient method that provably converges to second-order stationary point policies (SOSPs) for any differentiable policy classes. The proposed method is computationally efficient, and it judiciously uses cubic-regularized subroutines to escape saddle points while at the same time minimizing the Hessian-based computations. We prove that the method enjoys the sample complexity of O˜(ϵ−3.5), which improves upon the current optimal complexity O˜ϵ−4.5. Finally, experimental results are provided to demonstrate the effectiveness of the method.

A Subsampling Line-Search Method with Second-Order Results

In many contemporary optimization problems such as those arising in machine learning, it can be computationally challenging or even infeasible to evaluate an entire function or its derivatives. This motivates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees obtained through classical globalization techniques in optimization, such as a line search. Using subsampled function values is particularly challenging for the latter strategy, which relies upon multiple evaluations. For nonconvex data-related problems, such as training deep learning models, one aims at developing methods that converge to second-order stationary points quickly, that is, escape saddle points efficiently. This is particularly difficult to ensure when one only accesses subsampled approximations of the objective and its derivatives. In this paper, we describe a stochastic algorithm based on negative curvature and Newton-type directions that are computed for a subsampling model of the objective. A line-search technique is used to enforce suitable decrease for this model; for a sufficiently large sample, a similar amount of reduction holds for the true objective. We then present worst-case complexity guarantees for a notion of stationarity tailored to the subsampling context. Our analysis encompasses the deterministic regime and allows us to identify sampling requirements for second-order line-search paradigms. As we illustrate through real data experiments, these worst-case estimates need not be satisfied for our method to be competitive with first-order strategies in practice.

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.

ARES: a fast and accurate tool for the identification of plasma stationary points and separatrix

In this work, triangular C 1 interpolation based on Argyris element (AE) is used to develop the ARgyris element based searching (ARES) algorithm, an efficient tool for the accurate identification of stationary points (SPs), e.g. X-point, magnetic axis, snowflake point, etc and separatrix in magnetic confinement fusion application. AE-based interpolation allows the formulation of the nonlinear problem of SP identification on the triangle as a vectorial root-finding problem, addressed via different Newton-like algorithms. The proposed method is able to detect both first-order SP (e.g. X-points and magnetic axis, where the norm of the gradient vanishes) and second-order SP (e.g. snowflake points, where both the norm of the gradient and of the second derivatives vanishes). The separatrix is detected inside the triangles via the novel zero line searching algorithm, and is described by means of quintic spline interpolant of its components. Thanks to its modularity and efficiency, ARES is suitable to run on structured and unstructured triangular meshes, and can be readily integrated with any kind of Grad–Shafranov solver based on 2D triangular finite element method, or used for real-time SP and separatrix identification for control oriented purposes.

Quantum algorithms for escaping from saddle points

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a functionf:Rn→R, our quantum algorithm outputs anϵ-approximate second-order stationary point usingO~(log2⁡(n)/ϵ1.75)queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. withO~(log6⁡(n)/ϵ1.75)queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms oflog⁡nand matches its complexity in terms of1/ϵ. Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity withlog⁡nfactors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings.